On Free $ω$-Continuous and Regular Ordered Algebras

Abstract

We study varieties of certain ordered $Σ$-algebras with restricted completeness and continuity properties. We give a general characterization of their free algebras in terms of submonads of the monad of $Σ$-coterms. Varieties of this form are called \emph{quasi-regular}. For example, we show that if $E$ is a set of inequalities between finite $Σ$-terms, and if $\mathcal{V}_ω$ and $\mathcal{V}_\mathrm{reg}$ denote the varieties of all $ω$-continuous ordered $Σ$-algebras and regular ordered $Σ$-algebras satisfying $E$, respectively, then the free $\mathcal{V}_\mathrm{reg}$-algebra $F_\mathrm{reg}(X)$ on generators $X$ is the subalgebra of the corresponding free $\mathcal{V}_ω$-algebra $F_ω(X)$ determined by those elements of $F_ω(X)$ denoted by the regular $Σ$-coterms. This is a special case of a more general construction that applies to any quasi-regular family. Examples include the *-continuous Kleene algebras, context-free languages, $ω$-continuous semirings and $ω$-continuous idempotent semirings, OI-macro languages, and iteration theories.